02 - Kinematic & Motion Analysis

Our proposed mechanism has two main parts. First, it includes one link attached to the ground (foot) with a slider-crank that provides resistive feedback at an optimal position to prevent an excessive shank angle where the anterior knee exceeds the big toe (Figure 2). Second, it includes a set of linkages dedicated to provide proprioceptive feedback to the user when their anterior thigh is parallel to the ground.

As the person performing the motion progresses through the motion, the link attached to the shank (c) will slide up and hit the optimal position; any motion forward following will result in the spring deflecting, providing resistance to the knee (ensuring over flexion does not occur).

The kinematic challenges involved with our mechanism include:

1) Determining where the spring must be positioned to optimally provide resistive force against further motion such that the knee does not move past the user's big toe.

2) Determining the optimal linkage lengths for the thigh proprioceptive feedback.

Figure 2: Kinematic analysis and design diagram overview

To address the first kinematic challenge, the slider position can be calculated as a function of the ankle joint angle θ2 (Figure 3). The optimal value of θ2 can be determined by geometric positional analysis based on the user’s anthropometric data when the knee and longest toe are in line. Then, using the vector loop method, we can then solve for θ4, the angle of the linkage arm attached to the slider, and b, the distance the slider must travel. These equations allow us to calculate how long our shank links need to be and where the spring should be placed to hinder further motion. Using these equations, we tested different linkage lengths to see what provided the most optimal position by plotting them in MATLAB as seen in Figures 8 and 9. The plots provided us the visuals we needed to determine what would stop the knee (point A) from moving too far forward. The ankle joint angle and the slider-crank angle are plotted in Figure 7 (a) to show how much the slider-crank link must move in order to accommodate the inputted knee angle. And the horizontal displacement of knee and displacement of slider from ankle are plotted in Figure 7 (b) to show how the desired spring placement is determined. 

Figure 3: Positional analysis using the vector loop equation to solve for θ4 and b.

Addressing the second kinematic problem, we performed another vector loop equation analysis on a local x2y2 frame between links f, g, and h. This required a more difficult kinematic analysis because the thigh feedback linkages are coupled to the shank link, and it involved determining relative positions between the thigh and shank linkages. From this process, we could derive equations (2) and (3) as seen in Figure 4 which have four unknowns: f, g, h, and θ5.

Figure 4: Positional analysis using the vector loop equation to set up equations (2) and (3) for solving optimal lenghts of f and g.

In order to calculate the lengths of f and g, we had to first determine h and θ5. We used two known conditions, an optimal and neutral position, which provided us the parameters to determine the value of θ5 and h, reducing our unknowns to only f and g. Figure 5 below details each condition and shows how we solved for θ5 starting from knowing θ2 in the optimal position and how we solved for h in the neutral position. Figure 6 below provides the final two equations (5) and (6) which enabled us to define all geometry for our mechanism.

Figure 5: Two known conditions: optimal and neutral, which provide the necessary parameters to solve for link lengths of f and g.

Figure 6: Equations (5) and (6) used to calculate link lengths of f and g.